The origin of Poisson algebras lies in the Poisson geometry of the 1970s, and since then these algebras have shown their importance in several areas of mathematics and physics, such as Poisson manifolds, algebraic geometry, operads, quantization theory, quantum groups, and classical and quantum mechanics. The study of all possible Poisson algebra structures with fixed Lie or associative part is a popular topic in the theory of Poisson algebras [1]}, [2]}, [3]}, [4]}.
Recently, Bai, Bai, Guo, and Wu [5]} have introduced a dual notion of the Poisson algebra, called transposed Poisson algebra, by exchanging the roles of the two binary operations in the Leibniz rule defining the Poisson algebra.
They have shown that a transposed Poisson algebra defined this way not only shares common properties of a Poisson algebra, including the closedness under tensor products and the Koszul self-duality as an operad, but also admits a rich class of identities. More significantly, a transposed Poisson algebra naturally arises from a Novikov-Poisson algebra by taking the commutator Lie algebra of the Novikov algebra.
Thanks to [6]},
any unital transposed Poisson algebra is
a particular case of a “contact bracket” algebra [7]}
and a quasi-Poisson algebra [8]}.
Later, in a recent paper by Ferreira, Kaygorodov, and Lopatkin
a relation between \(\frac{1}{2}\) -derivations of Lie algebras and
transposed Poisson algebras have been established [9]}.
These ideas were used to describe all transposed Poisson structures
on Witt and Virasoro algebras in [9]};
on twisted Heisenberg-Virasoro, Schrodinger-Virasoro and
extended Schrodinger-Virasoro algebras in [11]};
on oscillator algebras in [6]}.
Fehlberg Júnior and Kaygorodov gave a way to construct new transposed Poisson algebras by the Kantor product of their multiplications in [13]}.
The \({\rm Hom}\) - and \({\rm BiHom}\) -versions of transposed Poisson algebras have been considered in [14]}, [15]}. A list of actual open questions on transposed Poisson algebras is given in [6]}.
